arXiv subjects
Publications and source records attributed to Bernd Ulrich.
We introduce several conjectures which mainly deal with restrictions on the Betti tables of licci ideals. We focus on a series of questions that compare the number of generators of homogeneous licci ideals in polynomial rings to the maximal last shift in their graded free resolution. We prove these conjectures in a large number of cases.
Given a scheme $X$ of finite type over the complex numbers, the Behrend function is a constructible function $\nu_X: X(\mathbb C) \rightarrow \mathbb Z $ introduced by Behrend in order to define enumerative invariants in Donaldson--Thomas theory. Even in simple cases, the Behrend function is very difficult to compute. In this article, we tackle the problem of computing the Behrend function of zero-dimensional schemes. We obtain a number of explicit formulas, in particular, for arbitrary zero-dimensional monomial schemes, thus providing vast generalizations of previous work of Graffeo--Ricolfi. Our main tools come from the theory of blowup algebras. Along the way, we establish results of independent interest related to the integer decomposition property, weighted Veronese subrings, and reduced fiber rings.
We prove a conjecture on the minimal number of generators of licci zero-dimensional ideals that are either monomial or have small Loewy colength.
We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher dimensions, prompting a search for sufficient conditions on the ideal. We show that if an integrally closed ideal contains a part of regular system of parameters of length $d-2$, where $d$ is the dimension of the regular local ring, then its Rees algebra is Cohen-Macaulay and normal. We also extend results of Goto and Ciuperc\u{a} by proving the same conclusion when the minimal number of generators of an ideal is at most $d+2$. Furthermore, we treat the case of integrally closed zero-dimensional ideals generated by $d+3$ homogeneous polynomials. Finally, using generic Bourbaki ideals, we generalize these results to integrally closed torsionfree modules of finite colength.
Let $(R,m,k)$ be a Golod ring. We show a recurrent formula for high syzygies of $k$ interms of previous ones. In the case of embedding dimension at most $2$, we provided complete descriptions of all indecomposable summands of all syzygies of $k$.
We study the behavior of multidegrees in families and the existence of numerical criteria to detect integral dependence. We show that mixed multiplicities of modules are upper semicontinuous functions when taking fibers and that projective degrees of rational maps are lower semicontinuous under specialization. We investigate various aspects of the polar multiplicities and Segre numbers of an ideal and introduce a new invariant that we call polar-Segre multiplicities. In terms of polar multiplicities and our new invariants, we provide a new integral dependence criterion for certain families of ideals. By giving specific examples, we show that the Segre numbers are the only invariants among the ones we consider that can detect integral dependence. Finally, we generalize the result of Gaffney and Gassler regarding the lexicographic upper semicontinuity of Segre numbers.
In this article, we study the generalized Poincare problem from the opposite perspective, by establishing lower bounds on the degree of the vector field in terms of invariants of the variety.
The core of an ideal is defined as the intersection of all of its reductions. In this paper we provide an explicit description for the core of a monomial ideal $I$ satisfying certain residual conditions, showing that ${\rm core}(I)$ coincides with the largest monomial ideal contained in a general reduction of $I$. We prove that the class of lex-segment ideals satisfies these residual conditions and study the core of lex-segment ideals generated in one degree. For monomial ideals that do not necessarily satisfy the residual conditions and that are generated in one degree, we conjecture an explicit formula for the core, and make progress towards this conjecture.
Schemes defined by residual intersections have been extensively studied in the case when they are Cohen-Macaulay, but this is a very restrictive condition. In this paper we make the first study of a class of natural examples far from satisfying this condition, the rank 1 loci of generic $2\times n$ matrices. Here we compute their depths and many other properties. These computations require a number of novel tools.
We provide a generalization of Jouanolou duality that is applicable to a plethora of situations. The environment where this generalized duality takes place is a new class of rings, that we introduce and call weakly Gorenstein. As a main consequence, we obtain a new general framework to investigate blowup algebras. We use our results to study and determine the defining equations of the Rees algebra of certain families of ideals.
If I is an ideal in a Gorenstein ring S and S/I is Cohen-Macaulay, then the same is true for any linked ideal I'. However, such statements hold for residual intersections of higher codimension only under very restrictive hypotheses, not satisfied even by ideals as simple as the ideal L_n of minors of a generic 2 x n matrix when n>3. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of I. For example, we prove that if K is the residual intersection of L_n by 2n-3 general quadratic forms in L_n, then S/K is integrally closed with isolated singularity and I^{n-3} S/K is a self-dual Maximal Cohen-Macaulay module over S/K with linear free resolution over S. The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented.
We prove that two arbitrary ideals $I \subset J$ in an equidimensional and universally catenary Noetherian local ring have the same integral closure if and only if they have the same multiplicity sequence. We also obtain a Principle of Specialization of Integral Dependence, which gives a condition for integral dependence in terms of the constancy of the multiplicity sequence in families.
Let R be a non-negatively graded Cohen-Macaulay ring with R_0 a Cohen-Macaulay factor ring of a local Gorenstein ring. Let d be the dimension of R, m be the maximal homogeneous ideal of R, and M be a finitely generated graded R-module. It has long been known how to read information about the socle degrees of the local cohomology module H_m^0(M) from the twists in position d in a resolution of M by free R-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of small Krull dimension. The present paper reads information about the maximal generator degree (rather than the socle degree) of H_m^0M from the twists in position d-1 (rather than position d) in an approximate resolution of M. We apply the local cohomology results to draw conclusions about the maximum generator degree of the second symbolic power of the prime ideal defining a monomial curve and the second symbolic power of the ideal defining a finite set of points in projective space. There is an application to general hyperplane sections of subschemes of projective space over an infinite field. There is an application of the local cohomology techniques to partial Castelnuovo-Mumford regularity. An application to the ideals generated by the lower order Pfaffians of an alternating matrix will appear in a future paper. One additional application to the study of blow-up algebras appears in a separate paper.
Paper has been accepted for publication in the Journal für die reine und Angewandte Mathematik. This version contains the corrections and additional references made in the Galley proofs.
Consider a rational projective plane curve C parameterized by three homogeneous forms h1,h2,h3 of the same degree d in the polynomial ring R=k[x,y] over the field k. Extracting a common factor, we may harmlessly assume that the ideal I=(h1,h2,h3)R has height two. Let phi be a homogeneous minimal Hilbert-Burch matrix for the row vector [h1,h2,h3]. So, phi is a 3 by 2 matrix of homogeneous forms from R; the entries in column m have degree dm, with d1 \le d2 and d1+d2=d. The Rees algebra $cal R$ of I is the subring k[h1t,h2t,h3t] of the polynomial ring k[t]. The bi-projective spectrum of $cal R$ is the graph of the parameterization of C; and therefore, there is a dictionary which translates between the singularities of C and the algebra structure of $cal R$. The ring $cal R$ is the quotient of the symmetric algebra Sym(I) by the ideal, A, of local cohomology with support in the homogeneous maximal ideal of R. The ideal A_{\ge d2-1}, which is an approximation of A, can be calculated using linkage. We exploit the bi-graded structure of Sym(I) in order to describe the structure of an improved approximation A_{\ge d1-1} when $d1<d2$ and phi has a generalized zero in its first column. (The later condition is equivalent to assuming that C has a singularity of multiplicity d2.) In particular, we give the bi-degrees of a minimal bi-homogeneous generating set for this ideal. When 2=d1<d2 and phi has a generalized zero in its first column, then we record explicit generators for A. When d1=d2, we provide a translation between the bi-degrees of a bi-homogeneous minimal generating set for A_{d1-2} and the number of singularities of multiplicity d1 which are on or infinitely near C. We conclude with a table which translates between the bi-degrees of a bi-homogeneous minimal generating set for A and the configuration of singularities of C in the case that the curve C has degree six.
We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An important step of the proof is the calculation of the degree of the variety parametrized by the forms generating the grade three Gorenstein ideal.
Let R be a standard graded polynomial ring in f variables over a field and Psi be an f by g matrix of linear forms from R, where g is positive and less than f. Assume that the row vector of variables annihilates Psi and that the ideal I generated by the g by g minors of Psi has grade exactly one short of the maximum possible grade. We resolve R/I, prove that I has a g-linear resolution, record explicit formulas for the h-vector and multiplicity of R/I, and prove that if f-g is even, then the ideal I is unmixed. Furthermore, if f-g is odd, then we identify an explicit generating set for the unmixed part, I^{unm}, of I, resolve R/I^{unm}, and record explicit formulas for the h-vector of R/I^{unm}. These results have applications to the study of the blow-up algebras associated to linearly presented grade three Gorenstein ideals.
Our object of study is a rational map Psi from projective s-1 space to projective n-1 space defined by homogeneous forms g1,...,gn, of the same degree d, in the homogeneous coordinate ring R=k[x1,...,xs] of projective s-1 space. Our goal is to relate properties of Psi, of the homogeneous coordinate ring A=k[g1,...,gn] of the variety parametrized by Psi, and of the Rees algebra R[It], the bihomogeneous coordinate ring of the graph of Psi. For a regular map Psi, for instance, we prove that R[It] satisfies Serre's condition R_i, for some positive i, if and only if A satisfies R_{i-1} and Psi is birational onto its image. Thus, in particular, Psi is birational onto its image if and only if R[It] satisfies R_1. Either condition has implications for the shape of the core, namely, the core of I is the multiplier ideal of I to the power s and the core of I equals the maximal homogeneous ideal of R to the power sd-s+1. Conversely, for s equal to two, either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of g1,...,gn, we give an explicit method to reduce the non-birational case to the birational one when s is equal to 2.